My position on science and religion/spirituality/supernatural stuffs is that they are compatible.
Let me define science first, I mean the scientific method of having to test hypothesis before accepting it as a model of how the world works. It doesn't include dogmatism of philosophical believes like the brain is the mind, or materialism/physicalism. It doesn't include the opinion of scientists, although this is largely how science progress, through the collective agreement of the scientists in the field.
I do not distinguish much between religion, spirituality and supernatural stuffs as Buddhism has them intertwined together, but at the same time it is scientific at heart, so these elements are not that crucial to Buddhism, the crucial thing is to be our own scientist of our mind to observe using mindfulness about how our mind works based on working hypothesis provided by the Buddha. The claim is, if done according to the instructions with all the right conditions brought about by practising, we will end all forms of dissatisfaction forever for each individual.
So by religion, I do not include other religions in general, mainly only referring to Buddhism, by spirituality, I do not include the new age stuffs, just the practise of meditation, wisdom and morality as found in Buddhism. By supernatural stuffs, I do not include all the claims by whatever nut case out there but mainly rebirth (of which there are sufficient objective evidence to be accepted as fact), kamma, beings of other realms and psychic powers via meditation.
Most scientists have mainly have a problem with the supernatural stuffs. And of course that religious people tend to be anti-science because science cannot support their supernatural claims. This attitude from religious people are indeed unwarranted and downright dangerous. If religion is meant to bring us to the truth, then religious people should like science as it did uncover many ignorance about the world. It is people who are climate denialist, anti-vaccine who are putting human lives in danger. (Not to mention some people believe in flat earth!)
The supernatural stuffs are mainly outside the current purview of scientific investigation mainly because it doesn't fit into the materialism philosophy. So scientists who adhere to that philosophy may become unjust in judging raw data or too quick in dismissing evidence and hypothesis outside of the materialism worldview.
Science requires the ability to reproduce things in a controlled environment to count as valid data. Often times this doesn't agree with some religious people out of respect for their gods or that we know too little about ghosts that we can't trap them like ghost busters to study them, leading to the widespread believe that ghost does not exist.
For those few who encountered these supernatural stuffs as they go deeper into spirituality and their religion, it becomes harder to communicate with the scientists because there can be a claim that they wish for their religion to be supported by science as their underlying motive for bringing up these data. Worse still is the reputation of their religion would be tarnished if for whatever reason the scientists dismiss the claim as false.
Most spiritual practitioners lack the training of scientists to be able to conduct experiments and gathering data for themselves should they obtain psychic powers via meditation. There can also be an issue of credibility and motivation. Most people who did not spend their life building themselves up as respected scientists might be ridiculed for publishing a paper on researching supernatural stuffs. Most people who can develop these psychic powers usually have no motivation to become a scientist but more motivated to attain to enlightenment. Those few who are scientists and great practitioners might see that it cannot really change people's mind anyway, those who believe place little value on scientific proof. Those who disbelieve might reject facts to upheld their precious materialism view. Only those who are in the middle of being open minded yet not so much faith may benefit from these findings.
And so pockets of humans are trapped in their own worldview blissfully ignorant of what the other thinks.
Many times it might not be worth doing the bridge as the negative repercussions are worse than the positive ones.
Having said that, this is one possible future about how science can proceed to make progress in discovering the supernatural laws. First, is through the meditation science. Once meditation is taught to the general populace an scientists until they can attain to the fourth jhana, and then proceed on to develop psychic powers, then those psychic powers will be new tools for scientific discovery into the supernatural.
Second, taking hints from the brahmajala sutta, merely having psychic powers like divine eye is not enough to uncover objective truths. Views cloud the interpretation of what is seen. Eg. Near death experience can be different based on the person's religion and believes. So experiments on having different and the same views will have to be conducted to find out the relationship between views and how people interprets their spiritual experience. Mindfulness of feelings and views will help a lot in uncovering the views and interpretation and how they form.
Lastly, in particular for Buddhism, it can be tested of having right view as described in Buddhism can lead to the end of all dissatisfaction as claimed by the Buddha. Naturally, as Buddhist, we will say that there is thousands of years of data and proof. But from the perspective of science, the development must go step by step to establish theories and tools for investigating the mind before it can be accepted as scientific facts.
In that era, anti-science promotion will only harm the acceptance of what science will have discovered.
Right now materialism dogmatism is strong amongst many scientists and it will be hard on the side of spiritual practitioners to come to science. It can only be bridged by scientists coming to spiritualism and test it out. Of course, if the above scenario ever happens, we might have religious crisis on our hands as people decide to renounce other religions and how would leaders of other religion act. Or ignorance can run deeper and alienate a lot of humanity from proper science.
Science can also be used in another way as had been done by communism, to destroy all religions because of the believe that humans created God and religions without having to investigate if religions has anything to offer towards the search for truth. Especially tragic is what happened to Buddhism in China due to communism. So militant atheists, do investigate first, not all religions has unfalsifiable God.